We analyze pairing of fermions in two dimensions for fully-gapped cases with broken parity (P) and time-reversal (T), especially cases in which the gap function is an orbital angular momentum (l) eigenstate, in particular l = −1 (p-wave, spinless or spin-triplet) and l = −2 (d-wave, spinsinglet). For l = 0, these fall into two phases, weak and strong pairing, which may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topological invariant. For the spinless p-wave case, the weak-pairing phase has a pair wavefunction that is asympototically the same as that in the Moore-Read (Pfaffian) quantum Hall state, and we argue that its other properties (edge states, quasihole and toroidal ground states) are also the same, indicating that nonabelian statistics is a generic property of such a paired phase. The strong-pairing phase is an abelian state, and the transition between the two phases involves a bulk Majorana fermion, the mass of which changes sign at the transition. For the d-wave case, we argue that the Haldane-Rezayi state is not the generic behavior of a phase but describes the asymptotics at the critical point between weak and strong pairing, and has gapless fermion excitations in the bulk. In this case the weak-pairing phase is an abelian phase which has been considered previously. In the p-wave case with an unbroken U(1) symmetry, which can be applied to the double layer quantum Hall problem, the weak-pairing phase has the properties of the 331 state, and with nonzero tunneling there is a transition to the Moore-Read phase. The effects of disorder on noninteracting quasiparticles are considered. The gapped phases survive, but there is an intermediate thermally-conducting phase in the spinless p-wave case, in which the quasiparticles are extended.

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A recent theory of a compressible Fermi-liquid like state at Landau level filling factors ν = 1/q or 1 − 1/q, q even, is reviewed, with emphasis on the basic physical concepts.

The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spinpolarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k + 1-body interactions, for all integers k ≥ 1. The remarkably simple wavefunctions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k = 2 case is the Pfaffian. For k ≥ 2, the quasiparticle excitations of these systems are expected to possess nonabelian statistics, like those of the Pfaffian. For k = 3, these ground states have large overlaps with the ground states of the (2-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors ν = 2 + 3/5, 2 + 2/5.

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We formulate a quasiparticle theory for a single hole in a quantum antiferromagnet in the limit that the Heisenberg exchange energy is much less than the hopping matrix element, J «t. We consider the ground state of the spins to be either a quantum Neel state or a d-wave resonatingvalence-bond (RVB}state. We show in a self-consistent perturbation theory that the hole spectrum is strongly renormalized by the interactions with spin excitations. The hole can be described by a narrow quasiparticle band located at an energy of order -t with a quasiparticle residue of order J/t and a bandwidth of order J. Above the quasiparticle band is an incoherent band of width of order t.Our results indicate that the energy scale for any coherent phenomenon involving the holes is 6J, where 5 is the doping concentration. In the Neel state we perform a spin-wave expansion on an anisotropic Heisenberg model. In the Ising limit we reproduce previously known results and then expand perturbatively about that limit. In this expansion we find that the holes have a quasiparticle residue of J,/t and a bandwidth of J~. In the Heisenberg limit we employ a "dominant pole" approximation in which we ignore contributions to the self-energy from the incoherent part of the hole spectrum. A similar technique is used to study the d-wave RVB state. The relevance of our results to recent optical experiments is discussed.